# The centered discrepancy D_y at finite x: what a census can and cannot see **No estimate for D_y is obtained and the sufficient input (16) of [moving-cutoff-parity.md](moving-cutoff-parity.md) remains OPEN.** This note records a finite computation with two pre-registered falsifiers, what it refuted (nothing), and the one thing it did establish: that at reachable x the object is dominated by the finite-size error of its classical companion term, so a larger census could not see the parity object either. ## 1. Prior data and the decision this run informed No published or tabulated values of D_y, of Lambda(n-2)mu(n) in progressions, or of the neighbouring shifted-prime and two-point Chowla sums exist above x=10^4 in the owning convention; the search record is [history/reviews-0907/01](history/reviews-0907/01-shifted-prime-data-search.md) and the convention row is in [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md). The decision: whether lane A of [RESEARCH-EXECUTION.md](RESEARCH-EXECUTION.md) should aim at an absolute, Bombieri--Vinogradov-type theorem for the sequence f(n)=Lambda(n-2)mu(n), which would supply (16) through (13), or must exploit sign cancellation across the moduli; and, cheaper, whether (16) is numerically false at accessible scales, which would end the centered route. ## 2. The objects, exactly as computed [centered-discrepancy-measurement.js](centered-discrepancy-measurement.js) takes x=2^j, J=(x/2,x], y=ceil(x^(12/25)), Q=floor(x/y), standard Lambda with prime powers, f(n)=Lambda(n-2)mu(n), M=sum_J f, and evaluates D_y(x) of (9) as the finite atomic Stieltjes sum D_y = sum_{e<=Q odd sqfree} mu(e) [ sum_{n>ey, e|n} f(n) log(e/n) - (1/phi(e)) sum_{n>max(x/2,ey)} f(n) log(e/n) ], with the endpoint a_e excluded; the first bracket summed over all e is acc1, the second is the density term P. It also evaluates T1, T2, E_pp, E_even of (4)--(5) and (10) from the same factor lists, S(x)=sum_J Lambda(n)Lambda(n-2), and the absolute sum W1=sum_e log(x/e) max_t |Delta_e(t)| of (13) with the max over a 64-point grid in t (a lower bound for W1; at j in {16,18,20} the exact max gives grid/exact 0.9747, 0.9718, 0.9661). Pre-registered before the first full run: F1, D_y/x < -4/25 at any j>=26 refutes (16) at that scale; F2, W1grid/x > 2/25 at every j>=26 and non-decreasing makes the absolute form implausible; F3, the residual r(x) of (12) must fall with j; F4, comparison with four random-sign draws (mu(n) replaced by +-1 on the same support, mu(e) kept). Controls: S=T1+T2+E_pp and T2=acc1+E_even exactly; a naive per-modulus implementation directly from (9) at j in {16,18,20}; draws' mean within 4 se of zero; shift 4 differs from shift 2. An adversarial code review before the bound run ([history/reviews-0907/02](history/reviews-0907/02-centered-discrepancy-script-review.md)) found the E_even column double-subtracting P and a vacuous F2 on short runs; both fixed, no headline number affected. The constant A2 in the thresholds line was corrected from a wrong value before the bound run, for the reason recorded in [shifted-prime-mobius-sums.md](shifted-prime-mobius-sums.md) ยง2. ## 3. Readings Every figure is from the OUTPUT block. j indexes x=2^j; 3948794345 support integers at j=38; 8223.5 s total on 8 workers. 1. **F1 does not fire.** For j>=26 the extreme values of D_y/x are -0.004283 (j=27), -0.004180 (j=33) and +0.002449 (j=31); at j=38 it is 0.001476. The threshold is -0.16. The centered route is not refuted at these scales. This is "not refuted", never "supported": (16) is an asymptotic statement. 2. **F2 does not fire, and the absolute form is noise-sized.** W1grid/x falls monotonically from 0.322294 (j=26) to 0.077347 (j=38), crossing 2/25 at the last row; the grid undercount is about 3 percent, so the true W1/x at j=38 is at the threshold. Real W1 is within 8 percent of the control at every j>=26 (0.077347 against 0.080797 at j=38) and the slopes of log W1 against log x over j>=26 are 0.827 real against 0.825 control. The absolute sum is the size a random-sign model gives, roughly x^(1/2) Q^(1/2) times logarithms, and shrinks relative to x for that reason alone. Consistent with the absolute form being true; no bearing on proving it. 3. **Real D_y is far above the control, and the identity says why.** The ratio |D_y|/(control rms) is 6.014, 12.849, 8.252, 7.703, 7.374, 8.580, 21.816 at j=32..38, and the slope of log|D_y| over j>=26 is 0.863 against the control's 0.544. Read alone this would look like arithmetic structure. The raw comparison is dominated by a classical term on these measured scales. The exact identities S=T1+T2+E_pp and T2=acc1+E_even, with D_y=acc1-P, give D_y/x = (S/x - C2) - (T1/x - C2) - P/x - E_pp/x - E_even/x, and the columns show every term but one is negligible: S/x-C2 is at most 1.8e-4 in magnitude for j>=30 (1.80e-4 at j=32, 9e-6 at j=38), P/x is below 9e-5 (it is (P+2C2M)/x, below 5e-7, minus 2C2M/x with M/x below 7e-5), E_pp/x is below 3e-5, E_even/x below 2e-9. So D_y/x equals -(T1/x - C2) to within 2e-4 at every j>=30 (the largest deviation is 1.84e-4 at j=30; it is below 1e-4 at j=32, 33, 34, 36, 37, 38): at j=38, 0.001476 against -(-0.001473); at j=33, -0.004180 against -(0.004208); at j=35, 0.001132 against -(-0.001286). T1 is the classical term of (6), Bombieri--Vinogradov plus the Mobius mean, whose error is O_A(x/log^A x) for every fixed A; its measured size on this range is the truncation of the Mobius partial sum of (8) at u=y, computable without primes (item 8; the earlier reading "comparable to x/log^2 x with oscillating sign" was a size comparison, 1/log^2(2^38)=1.4e-3, not an identification). The control has no such term: its D_y is a genuine random-sign sum of size about 1e-4 x at j=38. The real D_y's excess over the control is therefore the slow convergence of a proved classical asymptotic, and the parity object's own fluctuation, which is what a census was meant to look at, is not isolated by this raw comparison. It is isolated by subtracting that computable term (item 8). This is a finite diagnostic, not a theorem about every reachable scale. 4. **F3.** |r(x)| falls from 0.017153 (j=20) through 0.010170 (j=24), 0.003895 (j=29) to 0.001471 (j=38) with oscillating sign; it is the T1 error by item 3. The derivation is not flagged. The rate is not resolved (a power of 1/log x between 2 and 3 fits the range; not fitted). 5. **The centering removes almost nothing at finite x.** P/x is at most 1.5e-4 in magnitude for j>=26 and D_y/x agrees with acc1/x to 4e-6 at j=38, because M is at random-sign size (real M/x against control rms: -3.28e-6 against 6.44e-6 at j=38). The trivial bound M<=A2 x/2 that sets the -4/25 tolerance is loose by four orders here, as [shifted-prime-mobius-sums.md](shifted-prime-mobius-sums.md) records; a derivation still cannot use that. 6. **Controls.** exactAlgebra=true (both identities hold to at most 1.3e-5 absolute at j=38 on sums of size 1e11); naiveMatches=true (relative differences 9e-15, 7.5e-15, 6.1e-15 at j=16,18,20); ctrlMeanWithin4se=true; shift4Differs=true at j=20,24,28. M at j=24 is -18508.63 here and -18508.6301 in the other script's block. 7. **Not measured:** any estimate for D_y; the rate of (12)'s error. By item 8, a census of D_y at reachable x does separate the parity object from the classical convergence once the truncation term of (8) is subtracted, and the separated object is the size of the random-sign control at every j>=30; so a larger run would add rows but no decision, and should not be made. (Superseded wording, 2026-09-07 to 2026-09-12: "no census of D_y at reachable x can separate the parity object from the classical convergence"; wrong in scope.) 8. **The classical term's error is a computable truncation term (2026-09-12, return #171, pre-registered file 770da37e...).** T1 truncates its Mobius sum at d<=y, so its Bombieri--Vinogradov main term is T1_main(x,y)=(x/2) sum_{d<=y odd sqfree} sum_{b odd sqfree,[d,b^2]<=x} -mu(d)mu(b) log d/phi([d,b^2]), the finite form of (8), computable with no prime above max(y,sqrt x). With delta_trunc=T1_main-C2 x and B=T1-T1_main: delta_trunc/x against T1/x-C2 is -0.002490/-0.002591 (j=31), +0.001282/+0.001349 (32), +0.004279/+0.004208 (33), +0.001004/+0.000983 (34), -0.001347/-0.001286 (35), +0.001276/+0.001204 (36), +0.000536/+0.000455 (37), -0.001508/-0.001473 (38); at j=30, -0.000130/+0.000450. RMS(B/x)/RMS(T1/x-C2)=0.106 over j=30..38. The corrected object D_y+delta_trunc against the control sd of this note's own random-sign draws has median 0.60 and maximum 2.38 over j=30..38 (raw D_y: 7.06 and 35.1); at j=38 it is -3.3e-5 x against 4.2e-5 x. At j<=29 the residual B is within 1 sd of an exact independent-thinning control at every row. Files and recipe on return #171 (t1main.c, thin385.c, analysis.py). Measured; nothing asymptotic; (16) untouched. ## 4. What this changes Nothing in the proof status. For lane A: an absolute BV-type theorem for f(n) is consistent with the data and would suffice through (13); the difficulty is entirely in proving it for this sequence, not in whether it holds. For finite testing: the raw centered representation shows, at every reachable scale, the truncation error of the classical term; with that computable term subtracted (item 8) it shows an object of random-sign size, consistent with the absolute form being true and silent on proving it. The bounded negative recorded here is about the raw measurement, so it is not repeated; the corrected comparison needs no new run. ## 5. Cost and custody 8223.5 s wall on 8 workers (j=38 single-threaded for the last two hours), about 2.3 GB per worker at j=38. Two earlier launches were killed within 20 minutes for the review fixes and the constant. Retained artifact: [centered-discrepancy-measurement.json](centered-discrepancy-measurement.json). The naive per-modulus implementation is the independent check of (9); the shifted-prime script's M column is the cross-codebase check.